DOCSLIB.ORG

Tippe Top Inversion As a Dissipation-Induced Instability∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tippe Top Inversion As a Dissipation-Induced Instability∗ SIAM J. APPLIED DYNAMICAL SYSTEMS c 2004 Society for Industrial and Applied Mathematics Vol. 3, No. 3, pp. 352–377 Tippe Top Inversion as a Dissipation-Induced Instability∗ † ‡ § Nawaf M. Bou-Rabee , Jerrold E. Marsden , and Louis A. Romero Abstract. By treating tippe top inversion as a dissipation-induced instability, we explain tippe top inversion through a system we call the modified Maxwell–Bloch equations. We revisit previous work done on this problem and follow Or’s mathematical model [SIAM J. Appl. Math., 54 (1994), pp. 597–609]. A linear analysis of the equations of motion reveals that only the equilibrium points correspond to the inverted and noninverted states of the tippe top and that the modified Maxwell–Bloch equations describe the linear/spectral stability of these equilibria. We supply explicit criteria for the spectral stability of these states. A nonlinear global analysis based on energetics yields explicit criteria for the existence of a heteroclinic connection between the noninverted and inverted states of the tippe top. This criteria for the existence of a heteroclinic connection turns out to agree with the criteria for spectral stability of the inverted and noninverted states. Throughout the work we support the analysis with numerical evidence and include simulations to illustrate the nonlinear dynamics of the tippe top. Key words. tippe top inversion, dissipation-induced instability, constrained rotational motion, axisymmetric rigid body AMS subject classifications. 70E18, 34D23, 37J15, 37M05 DOI. 10.1137/030601351 1. Introduction. Tippe tops come in a variety of forms. The most common geometric form is a cylindrical stem attached to a truncated ball, as shown in Figure 1.1. On a flat surface, the tippe top will rest stably with its stem up. However, spun fast enough on its blunt end, the tippe top momentarily defies gravity, inverts, and spins on its stem until dissipation causes it to fall over. This spectacular sequence of events occurs because, and in spite of, dissipation. Tippe top inversion is an excellent example of a dissipation-induced instability. Tippe top inversion is described by a system we call the modified Maxwell–Bloch equations. These ∗Received by the editors October 22, 2003; accepted for publication (in revised form) by M. Golubitsky January 22, 2004; published electronically July 6, 2004. This work was performed by an employee of the U.S. Government or under U.S. Government contract. The U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. Copyright is owned by SIAM to the extent not limited by these rights. http://www.siam.org/journals/siads/3-3/60135.html †Applied and Computational Mathematics, Caltech, Pasadena, CA 91125 ([email protected]). The research of this author was supported by the U.S. DOE Computational Science Graduate Fellowship through grant DE-FG02- 97ER25308. ‡Control and Dynamical Systems, Caltech, Pasadena, CA 91125 ([email protected]). The research of this author was partially supported by the National Science Foundation. §Sandia National Laboratories, P.O. Box 5800, MS 1110, Albuquerque, NM 87185-1110 ([email protected]). The research of this author was supported by Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. 352 TIPPE TOP INVERSION 353 Figure 1.1. From left: a sketch of the noninverted and inverted states of the tippe top. equations are a generalization of a previously derived normal form describing dissipation- induced instabilities in the neighborhood of the 1:1 resonance [4]. We will show in section 2 that the modified Maxwell–Bloch equations are the normal form for rotationally symmetric, planar dynamical systems. A dissipation-induced instability describes a neutrally stable equilibrium becoming spec- trally (and hence nonlinearly) unstable with the addition of dissipation. Dissipation-induced instability itself has a long history, which goes back to Thomson and Tait; see [14]. In its modern form, dissipation-induced instability was shown both to be a general phenomenon for gyroscopically stabilized systems and to provide a sharp converse to the energy momen- tum stability method by Bloch et al. in [1] and [2]. We refer the reader to these papers for additional examples and basic theory. By considering tippe top inversion as a dissipation-induced instability, we observe pre- cisely how tippe top inversion relates to well-understood dissipation-induced instabilities in gyroscopic systems. History and literature. Tippe top inversion has been much investigated in the literature but never satisfactorily resolved. In what follows we survey a selection of theoretical results relevant to this paper. We refer the reader to the work of Cohen [5] and Or [11] for more comprehensive surveys of the literature. Analysis of the tippe top dates back to the last century with Routh’s commentary on the rising of tops. Routh’s simple physical analysis made it clear that dissipation was fundamental in understanding the physics of tippe top inversion [12]. In 1977, Cohen confirmed Routh’s physical analysis through numerical simulation [5]. Cohen modeled the tippe top as a holo- nomic ball with an inhomogeneous, axisymmetric mass distribution on a fixed plane. Thus, the ball’s center of mass was on its axis of symmetry, but not coincident with its geometric center. Tippe top inversion corresponded to the ball’s center of gravity moving above its ge- ometric center. Cohen’s mathematical model, derived using Newtonian mechanics, is written in terms of Euler angles and assumed a sliding friction law. In reality, tippe top inversion is a transient phenomenon because of dissipation. How- ever, by neglecting frictional torque, the spinning inverted state becomes a steady-state phe- nomenon. With a sliding friction law that neglects frictional torque, Cohen’s equations of motion were amenable to a linear-stability analysis about the spinning inverted and nonin- verted states, but he did not carry out such an analysis in his work. In this sense the fact that the model neglects frictional torque is a feature rather than a defect of the friction law. Cohen concluded that a sliding friction law based on Coulomb friction explains tippe top 354 NAWAF M. BOU-RABEE, JERROLD E. MARSDEN, AND LOUIS A. ROMERO inversion. Moreover, Cohen showed that the inviscid ball (no sliding friction) does not invert. Coulomb friction is proportional to the normal force exerted by the surface at the point of contact and opposes the motion of the tippe top at the point of contact. This nonlinear friction law drops out of a standard linear-stability analysis. In 1994, Or extended Cohen’s work by generalizing his friction law, performing a dimen- sional analysis of the equations of motion, writing the equations of motion in convenient coordinates, and exploring the linearized behavior of the ball. To further facilitate a linear- stability analysis, Or added viscous friction to Cohen’s friction law. The viscous friction is linearly related to the slip velocity at the point of contact. Or’s work showed that inversion could occur because of viscous or Coulomb friction. Or’s analysis of tippe top inversion by Coulomb friction and viscous friction demonstrated viscous friction flips the top more rapidly than Coulomb friction. His analysis mentioned first integrals in the no-slip problem, but did not use them to explain the global behavior of the tippe top. There was good reason to avoid a global analysis of the tippe top: the theoretical model of a ball did not accurately model the contact effects of the tippe top stem. However, the fundamental physics behind the global behavior of the tippe top was still reasonably described by the equations of motion for a ball with an eccentric center of mass, i.e., a gyroscopic, axisymmetric rigid body rising from a gravitationally favorable position to an unfavorable one because of dissipation. This observation motivated some aspects of the present study and possibly Ebenfeld’s thesis on tippe top inversion from a Lyapunov perspective. In 1995, Ebenfeld and Scheck applied a Lyapunov method to a dimensional version of Or’s mathematical model. Using the energy as a Lyapunov function, they analyzed the orbital sta- bility of the spinning, sliding ball. They showed that, starting with nonzero slip, the standing, spinning tippe top tends to manifolds of constant energy with no-slip and no tangential surface force. Moreover, they showed that without slip the tippe top cannot invert. They classified all possible asymptotic solutions as either tumbling (precession, spin, and no nutation) or rotating (pure spin) solutions. They provide criteria for determining the Lyapunov stability of these solutions [6]. We have applied some techniques of the present work to the related, but different, “rising egg” problem. (See Bou-Rabee, Marsden, and Romero [3].) In that paper we address the contributions of Moffatt [10] and Ruina [13] to that problem. Main goals of this paper. The main result of this paper asserts that the stability of the noninverted and inverted tippe top (cf. Figure 1.1) and the existence of a heteroclinic connec- tion between these states is completely described by the modified Maxwell–Bloch equations: a normal form for rotationally symmetric, planar dynamical systems. This result is important because (1) it shows how tippe top inversion relates to well-studied dissipation-induced insta- bilities in gyroscopic systems; (2) it integrates the linear and nonlinear analysis; and (3) it simplifies the analysis of tippe top inversion and yields explicit criteria for when the tippe top inverts. We use Or’s mathematical model with viscous friction to obtain equations of motion for the tippe top in a convenient set of coordinates.
Load more

Recommended publications
  • Properties of Numerical Solutions Versus the Dynamical Equations
    Dynamics of the Tippe Top – properties of numerical solutions versus the dynamical equations Stefan Rauch-Wojciechowski, Nils Rutstam [email protected], [email protected] Matematiska institutionen, Linköpings universitet, SE–581 83 Linköping, Sweden. October 19, 2018 Abstract We study the relationship between numerical solutions for inverting Tippe Top and the structure of the dynamical equations. The numerical solutions confirm oscillatory behaviour of the inclination angle θ(t) for the symmetry axis of the Tippe Top. They also reveal further fine features of the dynamics of inverting solutions defining the time of inversion. These features are partially understood on the basis of the underlying dynamical equations. Key words: tippe top; rigid body; nonholonomic mechanics; numerical solutions 1 Introduction A Tippe Top (TT) is modeled by an axially symmetric sphere of mass m and radius R with center of mass (CM) shifted w.r.t. the geometrical center O by Rα, 0 < α < 1 (see Fig. 1). In a toy TT it is achieved by cutting off a slice of a sphere and by substituting it with a small peg that is used for spinning the TT. The sphere is rolling and gliding on a flat surface and subjected to the gravitational force mgzˆ. arXiv:1306.5641v1 [math.DS] 24 Jun 2013 When spun slowly− on the spherical part the TT spins wobbly for some time and comes to a standstill due to loss of energy caused by spinning friction. However when the initial spin is sufficiently fast the TT displays a counterin- tuitive behaviour of flipping upside down to spin on the peg with CM above the geometrical center O.
    [Show full text]
  • Dynamics of Nano Tippe
    Dynamics of nano tippe top Yue Chan∗, Ngamta Thamwattana and James M. Hill Nanomechanics Group, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia October 26, 2018 Abstract We investigate the motion of a nano tippe top, which is formed from a C60 fullerene, and which is assumed to be spinning on either a graphene sheet or the interior of a single-walled carbon nanotube. We assume no specific geometric configuration for the top, however for example, the nano tippe top might be formed by joining a fullerene C60 with a small segment of a smaller radius carbon nanotube. We assume that it is spinning on a graphene sheet or a carbon nanotube surface only as a means of positioning and isolating the device, and the only effect of the graphene or the carbon nanotube surface is only through the frictional effect generated at the point of the contact. We employ the same basic physical ideas originating from the classical tippe top and find that the total retarding force, which comprises both a frictional force and a magnetic force at the contact point between the C60 fullerene and the graphene sheet or the inner surface of the single-walled carbon nanotube, induces the C60 molecule to spin and precess from a standing up position to a lying down position. Unlike the classical tippe top, the nanoscale tippe top does not flip over since the gravitational effect is not sufficient at the nano scale. After the precession, while the molecular top spins about its lying down axis, if we apply the opposite retarding magnetic force at the contact point, then the molecule will return to its standing up position.
    [Show full text]
  • The Influence of the Rolling Resistance Model on Tippe Top Inversion
    The influence of the rolling resistance model on tippe top inversion A. A. Kilin, E. N. Pivovarova Abstract In this paper, we analyze the effect which the choice of a fric- tion model has on tippe top inversion in the case where the resulting action of all dissipative forces is described not only by the force applied at the contact point, but also by the additional rolling resis- tance torque. We show that, depending on the friction model used, the system admits different first integrals. In particular, we give ex- amples of friction models where the Jellett integral, the Lagrange integral or the area integral is preserved. We examine in detail the case where the action of all dissipative forces reduces to the horizontal rolling resistance torque. For this case we find permanent rotations of the system and analyze their linear stability. Also, we show that for this friction model no inver- sion is observed. arXiv:2002.06335v2 [math.DS] 20 Feb 2020 Introduction In this paper we analyze the dynamics of the tippe top on a smooth plane in the presence of forces and torques of rolling resistance. Tippe top inver- sion has attracted the attention of researchers for the last decades [1–12]. Reference [1] discusses the development of a spherical prototype of the top which a capable of performing various modes of motion (in particular, complete or partial inversion) by changing the mass-geometric characteris- tics. Many studies from the last century [2–7] and from the early part of this century [8–12] gave mathematical explanations of tippe top inversion.
    [Show full text]
  • Motion of the Tippe Top: Gyroscopic Balance Condition and Stability
    YNU-HEPTh-05-102 July 2005 Motion of the Tippe Top Gyroscopic Balance Condition and Stability Takahiro UEDA∗, Ken SASAKI† and Shinsuke WATANABE‡ Dept. of Physics, Faculty of Engineering, Yokohama National University Yokohama 240-8501, JAPAN Abstract We reexamine a very classical problem, the spinning behavior of the tippe top on a horizontal table. The analysis is made for an eccentric sphere version of the tippe top, assuming a modified Coulomb law for the sliding friction, which is a continuous function of the slip velocity vP at the point of contact and vanishes at vP =0. We study the relevance of the gyroscopic balance condition (GBC), which was discovered to hold for a rapidly spinning hard-boiled egg by Moffatt and Shimomura, to the inversion phenomenon of the tippe top. We introduce a variable ξ so that ξ = 0 corresponds to the GBC and analyze the behavior of ξ. Contrary to the case of the spinning egg, the GBC for the tippe top is not fulfilled initially. But we find from simulation that for those tippe tops which will turn over, the GBC will soon be satisfied approximately. It is shown that the GBC and the geometry lead to the classification of tippe tops into three groups: The tippe tops of Group I never flip over however large a spin they are given. Those of Group II show a complete inversion and the tippe tops of Group III tend to turn over up to a certain inclination angle θf such that θf <π, when they are spun sufficiently rapidly. There exist three steady states for the spinning motion of the tippe top.
    [Show full text]
  • Physics of the Phitop® Kenneth Brecher, Boston University, Boston, MA Rod Cross, University of Sydney, Sydney, Australia
    Physics of the PhiTOP® Kenneth Brecher, Boston University, Boston, MA Rod Cross, University of Sydney, Sydney, Australia he PhiTOP® (or TOP®) is a physics toy designed G due to N is equal to MgR and it acts into the page along the not only to act as a spinning top but also to appeal to positive Y-axis through G. Intuition might predict that the the eye and to the scientifically curious mind.1 It is top should rotate about the Y-axis, and fall onto the horizontal Tcurrently made in two versions, one from solid aluminum surface instead of rising, as it would in Fig. 2 if the top was not and the other solid brass. Each top is highly polished, and spinning. Intuition is clearly wrong. Instead, the top rotates is elliptical in one cross section and circular in another. It is about the vertical axis like any other spinning top, an effect therefore a prolate ellipsoid or a spheroid, as indicated in Fig. known as precession. 1. Its name derives from the ratio of the lengths of the major to minor axes, which is equal to the golden mean = (1 + 5)/2 ~ 1.618. It functions in the same way as a spinning egg or a spin- ning football in that it rises on one end if Fig. 1 An upright PhiTOP® spinning on a curved it is initially mirror. Fig. 2. Geometry of a spinning PhiTOP®. F is the friction force set spinning into the page. at sufficient speed with its long axis horizontal. The center of mass rises in an unexpected and somewhat mysterious man- When the top is first set in motion, it is set spinning rapidly ner, an effect that it also shares with a tippe top.
    [Show full text]
  • Dynamics of the Tippe Top Via Routhian Reduction
    Dynamics of the Tippe Top via Routhian Reduction M.C. Ciocci1 and B. Langerock2 1 Department of Mathematics, Imperial College London, London SW7 2AZ, UK 2 Sint-Lucas school of Architecture, Hogeschool voor Wetenschap & Kunst, B-9000 Ghent, Belgium October 22, 2018 Abstract We consider a tippe top modeled as an eccentric sphere, spinning on a horizontal table and subject to a sliding friction. Ignoring translational effects, we show that the system is reducible using a Routhian reduction technique. The reduced system is a two dimensional system of second order differential equations, that allows an elegant and compact way to retrieve the classification of tippe tops in six groups according to the existence and stability type of the steady states. 1 Introduction The tippe top is a spinning top, consisting of a section of a sphere fitted with a short, cylindrical rod (the stem). One typically sets off the top by making it spin with the stem upwards, which we call, from now on the initial spin of the top. When the top is spun on a table, it will turn the stem down towards the table. When the stem touches the table, the top overturns and starts spinning on the stem. The overturning motion as we shall see is a transition from an unstable (relative) equilibrium to a stable one. Experimentally, it is known that such a transition only occurs when the spin speed exceeds a certain critical value. Let us make things more precise and first describe in some detail the model used throughout this paper. The tippe top is assumed to be a spherical rigid body with its center of mass ǫ off the geometrical center of the sphere with radius R.
    [Show full text]
  • Dynamics of a Spherical Tippe Top Or Disk Rod Cross
    European Journal of Physics PAPER Related content - Effects of rolling friction on a spinning coin Dynamics of a spherical tippe top or disk Rod Cross To cite this article: Rod Cross 2018 Eur. J. Phys. 39 035001 - Spinning eggs and ballerinas Rod Cross - Experimenting with a spinning disk Rod Cross View the article online for updates and enhancements. Recent citations - A hemispherical tippe top Rod Cross This content was downloaded from IP address 200.130.19.152 on 22/04/2021 at 19:35 European Journal of Physics Eur. J. Phys. 39 (2018) 035001 (10pp) https://doi.org/10.1088/1361-6404/aaa34f Dynamics of a spherical tippe top Rod Cross School of Physics, University of Sydney, Sydney, Australia E-mail: [email protected] Received 9 November 2017, revised 7 December 2017 Accepted for publication 20 December 2017 Published 9 March 2018 Abstract Experimental and theoretical results are presented concerning the inversion of a spherical tippe top. It was found that the top rises quickly while it is sliding and then more slowly when it starts rolling, in a manner similar to that observed previously with a spinning egg. As the top rises it rotates about the horizontal Y axis, an effect that is closely analogous to rotation of the top about the vertical Z axis. Both effects can be described in terms of precession about the respective axes. Steady precession about the Z axis arises from the normal reaction force in the Z direction, while precession about the Y axis arises from the friction force in the Y direction.
    [Show full text]
  • TIPPE TOP Consider a Spherical Top of Radius R, Except That Its Center of Mass Is Not at the Center of the Sphere, but Rather At
    TIPPE TOP YOSI AVRON Abstract. Tippe-top is a top a spherical top that flips on its head when spun fast enough. The reason for that is hidden in an interesting constant of motion. This implies that the top has less energy when standing on its head. Consider a spherical top of radius R, except that its center of mass is not at the center of the sphere, but rather at ~® away from the center of the sphere. The top is moving on top of a table and the normal to the table isz ˆ. If we assume that the sphere touches the table at a point, then the problem has an interesting constant of motion, namely, with L~ the angular momentum of the top about its center of mass: C = L~ ¢ (Rzˆ + ®) Indeed ˙ C˙ = L~ ¢ (Rzˆ + ~®) + L~ ¢ ®˙ Now, with F~ the force at the contact point, the equations of motion is ˙ L~ = (Rzˆ + ~®) £ F From this follows that the first term drops and so C˙ = L~ ¢ ®˙ Now, compute the right hand side in the frame of the top. In this frane ~® is aa fixed vector so ~®˙ = ~! £ ® is perpendicular to both ! and ®. However, since the top is symmetric L~ = I2 ~! + (I3 ¡ I2)(! ¢ ®ˆ)® ˆ L~ is in the plane of ! and ®. Hence C˙ = 0. From this conservation law it follows the tippe top has less energy when its center of mass is above the center of the sphere than when it is below it. Indeed, from the coservation law !a(R + ®) = !b(R ¡ ®) 2 or !a < !b. The kinetic energy is proportional to ! and the result follows.
    [Show full text]
  • A Constraint-Based Approach to Rigid Body Dynamics for Virtual Reality Applications
    A Constraint-Based Approach to Rigid Body Dynamics for Virtual Reality Applications J¨org Sauer Elmar Sch¨omer Daimler-Benz AG Universit¨at des Saarlandes Research and Technology Lehrstuhl Prof. Hotz P.O.Box 2360 Im Stadtwald D-89013 Ulm D-66123 Saarbruc¨ ken Germany Germany email: [email protected] email: [email protected] Abstract classified according to figure 1: The GALILEO-system is a developmental state-of-the-art rig- id body simulation tool with a strong bias to the simulation GALILEO of unilateral contacts for virtual reality applications. On Unilateral the one hand the system is aimed at closing the gap be- Bilateral Constraints tween the ‘paradigms of impulse-based simulation’ and of Constraints ‘constraint-based simulation’. On the other hand the cho- sen simulation techniques enable a balancing of the trade-off between the real-time demands of virtual environments (i.e. non-permanent permanent contacts 15-25 visualizations per second) and the degree of physi- contacts cal correctness of the simulation. The focus of this paper lies on the constraint-based simulation approach to three- dimensional multibody systems including a scalable friction SIMULATION Impulse-based Constraint-based model. This is only one of the two main components of CAPABILITY Simulation the GALILEO-software-module. A nonlinear complementar- ity problem (NCP) describes the equations of motion, the contact conditions of the objects and the Coulomb friction Figure 1: The classification of contact situations. model. Further on we show, as an interesting evaluation ex- ample from the field of ‘classical mechanics’, the first rigid In the GALILEO-module (that is part of the VR-software- body simulation of the tippe-top.
    [Show full text]
  • Study of Equations for Tippe Top and Related Rigid Bodies
    Linköping Studies in Science and Technology. Thesis No. 1452 Study of equations for Tippe Top and related rigid bodies Nils Rutstam Department of Mathematics Linköping University, SE–581 83 Linköping, Sweden Linköping 2010 Linköping Studies in Science and Technology. Thesis No. 1452 Study of equations for Tippe Top and related rigid bodies Nils Rutstam [email protected] www.mai.liu.se Division of Applied Mathematics Department of Mathematics Linköping University SE–581 83 Linköping Sweden ISBN 978-91-7393-298-1 ISSN 0280-7971 LIU-TEK-LIC-2010:23 Copyright © 2010 Nils Rutstam Printed by LiU-Tryck, Linköping, Sweden 2010 Abstract The Tippe Top consist of a small truncated sphere with a peg as a handle. When it is spun fast enough on its spherical part it starts to turn upside down and ends up spinning on the peg. This counterintuitive behaviour, called inversion, is a curious feature of this dynamical system that has been studied for some time, but obtaining a complete description of the dynamics of inversion has proved to be a difficult problem. The existing results are either numerical simulations of the equations of mo- tion or asymptotic analysis that shows that the inverted position is the only attractive and stable position under certain conditions. This thesis will present methods to analyze the equations of motion of the Tippe Top, which we study in three equivalent forms that each helps us to un- derstand different aspects of the inversion phenomenon. Our study of the Tippe Top also focuses on the role of the underlying as- sumptions in the standard model for the external force, and what consequences these assumptions have, in particular for the asymptotic cases.
    [Show full text]
  • Constants of the Motion for Nonslipping Tippe Tops and Other Tops with Round Pegs C
    Constants of the motion for nonslipping tippe tops and other tops with round pegs C. G. Graya) and B. G. Nickelb) Department of Physics, University of Guelph, Guelph, Ontario NIG 2W1, Canada ͑Received 5 August 1999; accepted 21 December 1999͒ New and more illuminating derivations are given for the three constants of the motion of a nonsliding tippe top and other symmetric tops with a spherical peg in contact with a horizontal plane. Some rigorous conclusions about the motion can be drawn immediately from these constants. It is shown that the system is integrable, and provides a valuable pedagogical example of such systems. The equation for the tipping rate is reduced to one-dimensional form. The question of sliding versus nonsliding is considered. A careful literature study of the work over the past century on this problem has been done. The classic work of Routh has been rescued from obscurity, and some misstatements in the literature are corrected. © 2000 American Association of Physics Teachers. I. INTRODUCTION shows that there are three constants of the motion: the en- ergy, the so-called Jellett constant ͑a linear combination of The behavior of the tippe top has fascinated physicists for angular momentum components͒, and what we term the over a century.1–30 Textbook discussions are, however, still Routh constant. The physical significance of the Routh con- quite rare and brief.26,28,29 Figure 1 shows schematically the stant has not yet been understood, but we show that it in- small wooden commercial model readily available. ͑Appen- volves purely kinetic energy. dix A gives the specifications of the model in our posses- Our contribution is to simplify the derivation of the con- sion.͒ This type was introduced in Denmark a half century stants of the motion, and to discuss their physical signifi- ago.
    [Show full text]
  • Analysis of Dynamics of the Tippe Top
    Linköping Studies in Science and Technology. Dissertations, No. 1500 Analysis of Dynamics of the Tippe Top Nils Rutstam Department of Mathematics Linköping University, SE–581 83 Linköping, Sweden Linköping 2013 Linköping Studies in Science and Technology. Dissertations, No. 1500 Analysis of Dynamics of the Tippe Top Nils Rutstam [email protected] www.mai.liu.se Division of Applied Mathematics Department of Mathematics Linköping University SE–581 83 Linköping Sweden ISBN 978-91-7519-692-3 ISSN 0345-7524 Copyright © 2013 Nils Rutstam Printed by LiU-Tryck, Linköping, Sweden 2013 iii They have proven quite effectively that bumblebees indeed can fly against the field’s authority. -Einstürzende Neubauten Abstract The Tippe Top is a toy that has the form of a truncated sphere with a small peg. When spun on its spherical part on a flat supporting surface it will start to turn upside down to spin on its peg. This counterintuitive phenomenon, called in- version, has been studied for some time, but obtaining a complete description of the dynamics of inversion has proven to be a difficult problem. This is because even the most simplified model for the rolling and gliding Tippe Top is a non- integrable, nonlinear dynamical system with at least 6 degrees of freedom. The existing results are based on numerical simulations of the equations of motion or an asymptotic analysis showing that the inverted position is the only asymptot- ically attractive and stable position for the Tippe Top under certain conditions. The question of describing dynamics of inverting solutions remained rather in- tact. In this thesis we develop methods for analysing equations of motion of the Tippe Top and present conditions for oscillatory behaviour of inverting solu- tions.
    [Show full text]